proof that Euler φ function is multiplicative

Suppose that t=m⁢n where m,n are coprime. The Chinese remainder theorem (http://planetmath.org/ChineseRemainderTheorem) states that gcd⁡(a,t)=1 if and only if gcd⁡(a,m)=1 and gcd⁡(a,n)=1.

In other words, there is a bijective correspondence between these two sets:

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{a:a≡1(modt)}

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{a:a≡1(modm) and a≡1(modn)}

Now the number of positive integers not greater than t and coprime with t is precisely φ⁢(t), but it is also the number of pairs (u,v), where u not greater than m and coprime with m, and v not greater than n and coprime with n. Thus, φ⁢(m⁢n)=φ⁢(m)⁢φ⁢(n).